The Held—Karp algorithm and degree-constrained minimum 1-trees Authors Short Communication

Received: 01 August 1977 Revised: 08 May 1978 DOI :
10.1007/BF01609023

Cite this article as: Yamamoto, Y. Mathematical Programming (1978) 15: 228. doi:10.1007/BF01609023
Abstract
In this note we propose to find a degree-constrained minimum 1-tree in the Held—Karp algorithm [5, 6] for the symmetric traveling-salesman problem, and show that it is reduced to finding a minimum common basis of two matroids.

Key words
Traveling-Salesman Problem
1-tree
Matroid

References [1]

J. Edmonds, “Matroids and the greedy algorithm”,Mathematical Programming 1 (1971) 127–136.

[2]

S. Fujishige, “A primal approach to the independent assignment problem”,Journal of the Operations Research Society of Japan 20 (1977) 1–15.

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F. Glover and D. Klingman, “Finding minimum spanning trees with a fixed number of links at a node”, in:Combinatorial programming methods and applications (D. Reidel, Dordrecht, 1975) pp. 191–201.

[4]

C. Greene and T.L. Magnanti, “Some abstract pivot algorithms”,SIAM Journal on Applied Mathematics 29 (1975) 530–539.

[5]

M. Held and R.M. Karp, “The traveling-salesman problem and minimum spanning trees”,Operations Research 18 (1970) 1138–1162.

[6]

M. Held and R.M. Karp, “The traveling-salesman problem and minimum spanning trees: part II”,Mathematical Programming 1 (1971) 6–25.

[7]

M. Iri and N. Tomizawa, “An algorithm for finding an optimal ‘independent’ assignment”,Journal of the Operations Research Society of Japan 19 (1976) 32–57.

[8]

J.B. Kruskal, “On the shortest spanning subtree of a graph and the traveling-salesman problem”,Proceedings of the American Mathematical Society 2 (1956) 48–50.

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[10]

D.J.A. Welsh,Matroid theory (Academic Press, London, 1976).

© The Mathematical Programming Society 1978